Ok, I know what went wrong. I got confused by the notation.

Lets rename x --> x1 and y --> x2

from f(x1,x2) = (x1 + x2,x1 - x2) follow 2 equations:

(I) y1 = x1 + x2 = x1 - x2 + 2*x2 ( insert in (II))

(II) y2 = x1 - x2

--> y2 = y1 - 2*x2

--> x2 = 1/2*(y1 - y2) (insert in (I))

--> y1 = x1 + 1/2(y1 - y2)

--> x1 = 1/2*(y1 + y2)

Thus, both x1 and x2 are related to y1 and y2 only and the inverse function becomes:

f^(-1)(y1,y2) = 1/2*(y1 + y2, y1- y2)

I think mathematicians like it better is the argument in f() is named x so one renames

f^(-1)(x1,x2) = 1/2*(x1 + x2, x1 - x2)

cool